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Relationship between trace and phase space

  1. Jun 18, 2013 #1
    So I noticed we can define entropy in two very different ways:
    1) quantum mechanically
    [itex]S = -k Tr(\hat{\rho}\ln{(\hat{\rho})}) [/itex]
    2) classically
    [itex]S = -k \int \rho \ln{(\rho)} d\Gamma[/itex]

    where Tr is the trace and [itex]d\Gamma = \frac{1}{h^{3N}N!}\prod_{i}^{N} d^{3N}q_{i}d^{3N}p_{i}[/itex] is the phase space volume element.

    My question is how summing over the diagonal of an operator is like integrating the same quantity over phase space.

    Edit: I realise that we are working out a mean value.
    Last edited: Jun 18, 2013
  2. jcsd
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